MAJORIZING MEASURES WITHOUT MEASURES. By Michel Talagrand URA 754 AU CNRS

Size: px
Start display at page:

Download "MAJORIZING MEASURES WITHOUT MEASURES. By Michel Talagrand URA 754 AU CNRS"

Transcription

1 The Annals of Probability 2001, Vol. 29, No. 1, MAJORIZING MEASURES WITHOUT MEASURES By Michel Talagrand URA 754 AU CNRS We give a reformulation of majorizing measures that does not involve measures, but rather special sequences of partitions. This formulation is more convenient to perform chaining with different distances. 1. Introduction. The theory of majorizing measures seems the appropriate tool to estimate E sup t T X t, where X t t T is a stochastic process i.e., a family of r.v. s indexed by a set T such that one has some control over the tails of the increments X s X t for s t T. A typical such control would be of the form 1.1 u 0 PX s X t u 2exp u2 ds t 2 where d is a distance on T sub-gaussian processes. The theory of majorizing measures is a method to measure the size of a metric space T d in a manner appropriate to take full advantage of 1.1. This theory is explained in detail in [8]. The case 1.1 is very important, because of Gaussian processes. It is unfortunately rather exceptional that one can control the left-hand side of 1.1 in a sharpway as a function of one single distance on T. There are many cases e.g., infinitely divisible processes where this is best done using families of distances. The theory of majorizing measures has been extended to this setting. There does not exist a unified exposition of these results, which are scattered among a number of papers [5, 6] and [7]. This is unfortunate, because this extension of the theory of majorizing measures to the case of families of distances requires new ideas. This paper is motivated by the observation that one of these ideas can be expressed in the familar one distance setting and that this seems to have at least pedagogical interest. The present paper is therefore a complement to the expository paper [8]. Before we state our results, we recall the traditional definition of majorizing measures. Given α>0 and a metric space T d, we define 1.2 γ α T =γ α T d =inf sup log dε t T 0 µbt ε where Bt ε is the closed ball of center t and radius ε and where the infimum is taken over all the discrete probability measures on T. Several pages of [8] attempt to take some of the mystery out of this definition and to explain the meaning of this mysterious probability µ. The formulation we will give here offers the advantage of dispensing with µ. Received January 2000; revised May AMS 2000 subject classifications. 60G15, 60G17, 60G60. Key words and phrases. chaining, increment condition, bounded sample path. 411

2 412 M. TALAGRAND 1.3 We define γ α T =inf sup dt C k 2 k/α t T where the infimum is taken over all choices of finite subsets C k of T with card C 0 = 1, card C k 2 2k for k 1, and where, of course, dt C k is the distance from t to C k. Theorem 1.1. There exists a constant Kα depending only on α such that K α 1 γ α T γ αt Kαγ α T There are two rather distinct aspects to the theory of majorizing measures. One is, knowing the existence of a majorizing measure, to control processes such as in 1.1. We will show that the formulation 1.3 is well adapted to that purpose. The second topic, which is typically much harder, concerns construction of majorizing measures. Tools have been developed to that purpose, but the bottom line remains that one has to guess what the right choice of µ is. With the formulation 1.3, this amounts to guessing what sets one should take for C k. It is conceivable that, at least in certain situations, it is easier to guess these sets rather than to guess µ. The present paper was motivated by discussions with K. Ball about the case of the simplex, which is as follows. Consider the basis e 1 e n or n, provided with the Euclidean distance, and denote by T the convex hull of e 1 e n. It is known from the theory of Gaussian processes that γ 2 T is of order log n + 1, and it is an interesting exercise to prove this directly. A solution to this exercise is to be found in [8]. It is another interesting exercise left here as a challenge to the motivated reader to define sets C k that show that γ 2 T is at most of order log n + 1. Another important theme of majorizing measures is that of ultrametricity. For a metric space T d, consider an increasing sequence of finite partitions k of T. We assume 1.4 card 0 = 1 card k 2 2k For a point t in T, we denote by A k t the unique element of k that contains t. We denote by DA the diameter of a set A. Weset γ α 1.5 T =inf sup 2 k/α DA k t t T The infimum is taken over all choices of the sequence k. It should be obvious that γ α T γ αt, and the next result shows that, in fact, these two quantities are of the same order. 1.6 Theorem 1.2. There exists a constant Kα depending on α only such that Kα 1 γ α T γ αt Kαγ α T

3 MAJORIZING MEASURES 413 To illustrate the fact that the previous formulation is well adapted to the proof of upper bounds, we will prove the following. Theorem 1.3. Consider a set T provided with two distances d 1 d 2 anda process X t t T such that, for each u>0. u PX s X t u 2exp min d 1 s t u d 2 2s t Then E sup X t X s Cγ 1 T d 1 +γ 2 T d 2 s t T where C is universal. There are all kinds of possible variations different powers, etc. with the same proof. While proofs of upper bounds are overall easy, there seems to be a genuine difficulty in the proof of Theorem 1.3. In fact, the proof given in [1], page 327, contains a serious error. This gap was apparently first discovered by the referee of [2] as explained there. In the interval between the publication of [1] and this observation, a correct proof was given in [5] and [7] using the framework of the theory of majorizing measures with a family of distances. Even though Theorem 1.3 follows by combining the results of [5] and [7], the theory of majorizing measures for families of distances is a bit impressive at first sight, and the reader might enjoy the simple proof given here. Theorem 1.3 has been applied in [3] and [4]. 2. Proofs. We denote by K a constant depending on α only, not necessarily the same at each occurrence. Lemma 2.1. We have γ α T Kγ α T. Proof. Consider sets C k with card C 0 = 1, card C k 2 2k such that 2.1 t T dt C k 2 k/α 2γ α T Consider a probability measure µ on T such that 2.2 x C k µx 2 k 1 2 2k We will prove that 2.3 t T log dε Kγ α 0 µbt ε T and this will prove the lemma.

4 414 M. TALAGRAND We fix t, and we set Thus, for k 0, d k = min l k dt C l ε d k µbt ε 2 k 1 2 2k 2 2k+1 so, for k 1, dk log dε Kd d k µbt ε k 12 k/α Since, obviously, µbt ε = 1 for ε DT [where DT is the diameter of T], we have 2.5 log dε KDT d 0 µbt ε Now C 0 consists of one single point, and, by 2.2, t T dt C 0 2γ α T so that DT 4γ α T Summation of the relations 2.4 and 2.5 yields the result. Since, obviously, γ α following. γ α, to prove Theorem 1.2, it suffices to prove the Theorem 2.2. We have γ α T Kγ αt. Proof. We fix a probability measure µ on T such that 2.6 t T log dε 2γ 0 µbt ε α T Given k 1, we proceed to the basic construction. We set D = DT. Forl 0, we consider the set 2.7 C l = { t T µbt 2 l D 2 l+1 2 2k } Observe that C 0 = T. Consider Cl = C l C m m>l The sets C l are disjoint, and since C 0 = T, form a partition of T. Moreover, 2.8 t C l µbt 2 l 1 D 2 l+2 2 2k

5 MAJORIZING MEASURES 415 because t C l+1. Consider a subset N l of C l condition that is maximal subject to the s t N l s t ds t > 2 l+1 D The balls Bt 2 l D, for t N l, are disjoint, so that, by 2.7, 2.9 cardn l 2 l 1 2 2k Moreover, the maximality of N l implies that the balls Bt 2 l+1 D, for t N l, cover C l. Thus, we can partition C l in at most card N l sets of diameter at most 2 l+2 D. We now partition T as follows. First, we partition T into the sets C l. Then we partition each C l as explained. The resulting partition B k contains at most 2 2k 2 l 1 2 2k l 0 sets. To each t k, we associate the index lk t, defined by t C lk t.wehave 2.10 DB k t 2 lk t+2 D and, from 2.8, 2.11 µbt 2 lk t 1 D 2 lk t+2 2 2k We set 0 = 1 =T, and for k 2, we consider the partition k generated by B 1 B k 1, so that card k 2 2k, and the sequence k increases. Let us fix t T. We have, since A k t B k 1 t, DA k t2 k/α KD + DB k 1 t2 k/α k KD k/α 2 lk t k 1 using Consider the set and for l L, define L =l k l = lk t kl =maxk l = lk t It should be obvious that 2 k/α 2 lk t K 2 l 2 kl/α k 1 l L Now, for l L k = kl, 2.11 reads as µbt 2 l 1 D 2 l+2 2 2kl

6 416 M. TALAGRAND and thus l 1 D 2 l 2 D log dε µbt ε 2 l 2 D logmax1 2 2kl l 2 1/α 1 K 2 l D2 kl l 2 + 1/α 1 K 2 l D2 kl/α Kl + 2 1/α using the fact that, for a b > 0 a b + 1/α 1/Ka 1/α Kb 1/α. Summing the inequalities 2.13 and combining with 2.6 and 2.12 gives DA k t2 k/α KD + γ α T Kγ α T Since t was arbitrary, the proof is complete. Theorem 1.1 is a consequence of Lemma 2.1 and Theorem 2.3. Proof of Theorem 1.3. For j = 1 2, we consider an increasing sequence j k of partitions of T such that card j j 0 = 1 card k and 22k t T D j A j k t 2 k/j Kγ j T where D j A denotes the diameter of A for d j. This is possible by Theorem 1.2. We consider the partition k generated by k 1 2 and k, so that card 0 = 1, card k 2 2k+1 and 2.14 t T D 1 A k t2 k + D 2 A k t2 k/2 Kγ 1 T+γ 2 T All the difficulty of dealing with different distances is gone now that we have the correct normalization, and the rest is standard. For k 0, we consider an arbitrary set B k that contains exactly one point in each set of k. Consider a parameter v 1. For each k 0x B k+1 y B k, consider the event Thus, by 1.7, we have x y v X x X y v2 k d 1 x y+2 k/2 d 2 x y Px y v 2 exp v2 k

7 MAJORIZING MEASURES 417 Thus, if we denote by v the union of the events x y v over all choices of x y, wehave 2.15 Pv 2 2k+2 exp v2 k exp v 2 for v large enough. Next, if v does not occur, we bound sup t X t X t0, where B 0 =t 0. Given t in T, for k 1, let π k t =B k A k t. Since x y v does not occur, we have, since π k tπ k+1 t A k t, 2.16 X πk+1 t X πk t v2 k D 1 A k t + 2 k/2 D 2 A k t Summation of the relations 2.16 for k 0, together with 2.14, shows that X t X t0 Kvγ 1 T+γ 2 T Combining this with 2.15, we see that P sup X t X t0 Kvγ 1 T+γ 2 T exp v t 2 which implies the result. The extension of Theorem 1.3 to situations involving several distances is immediate. Such an extension is given in [2]. REFERENCES [1] Ledoux, M. and Talagrand, M Probability in Banach Spaces. Springer, Berlin. [2] Marcus, M A sufficient condition for the continuity of high order Gaussian chaos processes: high dimensional probability. Progr. Probab [3] Marcus, M. and Rosen, J Gaussian chaos and sample path properties of additive functionals of symmetric Markov processes. Ann. Probab , [4] Marcus, M. and Talagrand, M Continuity conditions for a class of Gaussian chaos processes related to continuous additive functionals of Lévy processes. J. Probab [5] Talagrand, M Regularity of infinitely divisible processes. Ann. Probab [6] Talagrand, M Construction of majorizing measures, Bernoulli processes and cotype. Geom. Funct. Anal [7] Talagrand, M The supremum of certain canonical processes. Amer. J. Math [8] Talagrand, M Majorizing measures: the generic chaining. Ann. Probab Equipe d Analyse URA 754 au CNRS 4 Place Jussieu Boite Paris Cedex 05 France

The Rademacher Cotype of Operators from l N

The Rademacher Cotype of Operators from l N The Rademacher Cotype of Operators from l N SJ Montgomery-Smith Department of Mathematics, University of Missouri, Columbia, MO 65 M Talagrand Department of Mathematics, The Ohio State University, 3 W

More information

Introduction and Preliminaries

Introduction and Preliminaries Chapter 1 Introduction and Preliminaries This chapter serves two purposes. The first purpose is to prepare the readers for the more systematic development in later chapters of methods of real analysis

More information

The small ball property in Banach spaces (quantitative results)

The small ball property in Banach spaces (quantitative results) The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence

More information

Weak and strong moments of l r -norms of log-concave vectors

Weak and strong moments of l r -norms of log-concave vectors Weak and strong moments of l r -norms of log-concave vectors Rafał Latała based on the joint work with Marta Strzelecka) University of Warsaw Minneapolis, April 14 2015 Log-concave measures/vectors A measure

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES

SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES ARCHIVUM MATHEMATICUM (BRNO) Tomus 42 (2006), 167 174 SOME PROPERTIES ON THE CLOSED SUBSETS IN BANACH SPACES ABDELHAKIM MAADEN AND ABDELKADER STOUTI Abstract. It is shown that under natural assumptions,

More information

GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n

GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. A. ZVAVITCH Abstract. In this paper we give a solution for the Gaussian version of the Busemann-Petty problem with additional

More information

Estimates for probabilities of independent events and infinite series

Estimates for probabilities of independent events and infinite series Estimates for probabilities of independent events and infinite series Jürgen Grahl and Shahar evo September 9, 06 arxiv:609.0894v [math.pr] 8 Sep 06 Abstract This paper deals with finite or infinite sequences

More information

CHAPTER 7. Connectedness

CHAPTER 7. Connectedness CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

More information

A REPRESENTATION FOR THE KANTOROVICH RUBINSTEIN DISTANCE DEFINED BY THE CAMERON MARTIN NORM OF A GAUSSIAN MEASURE ON A BANACH SPACE

A REPRESENTATION FOR THE KANTOROVICH RUBINSTEIN DISTANCE DEFINED BY THE CAMERON MARTIN NORM OF A GAUSSIAN MEASURE ON A BANACH SPACE Theory of Stochastic Processes Vol. 21 (37), no. 2, 2016, pp. 84 90 G. V. RIABOV A REPRESENTATION FOR THE KANTOROVICH RUBINSTEIN DISTANCE DEFINED BY THE CAMERON MARTIN NORM OF A GAUSSIAN MEASURE ON A BANACH

More information

LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS. S. G. Bobkov and F. L. Nazarov. September 25, 2011

LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS. S. G. Bobkov and F. L. Nazarov. September 25, 2011 LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS S. G. Bobkov and F. L. Nazarov September 25, 20 Abstract We study large deviations of linear functionals on an isotropic

More information

Math 320-2: Midterm 2 Practice Solutions Northwestern University, Winter 2015

Math 320-2: Midterm 2 Practice Solutions Northwestern University, Winter 2015 Math 30-: Midterm Practice Solutions Northwestern University, Winter 015 1. Give an example of each of the following. No justification is needed. (a) A metric on R with respect to which R is bounded. (b)

More information

Extensions of Lipschitz functions and Grothendieck s bounded approximation property

Extensions of Lipschitz functions and Grothendieck s bounded approximation property North-Western European Journal of Mathematics Extensions of Lipschitz functions and Grothendieck s bounded approximation property Gilles Godefroy 1 Received: January 29, 2015/Accepted: March 6, 2015/Online:

More information

REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS. Julien Frontisi

REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS. Julien Frontisi Serdica Math. J. 22 (1996), 33-38 REPRESENTABLE BANACH SPACES AND UNIFORMLY GÂTEAUX-SMOOTH NORMS Julien Frontisi Communicated by G. Godefroy Abstract. It is proved that a representable non-separable Banach

More information

ON A MEASURE THEORETIC AREA FORMULA

ON A MEASURE THEORETIC AREA FORMULA ON A MEASURE THEORETIC AREA FORMULA VALENTINO MAGNANI Abstract. We review some classical differentiation theorems for measures, showing how they can be turned into an integral representation of a Borel

More information

Fragmentability and σ-fragmentability

Fragmentability and σ-fragmentability F U N D A M E N T A MATHEMATICAE 143 (1993) Fragmentability and σ-fragmentability by J. E. J a y n e (London), I. N a m i o k a (Seattle) and C. A. R o g e r s (London) Abstract. Recent work has studied

More information

Stat 451: Solutions to Assignment #1

Stat 451: Solutions to Assignment #1 Stat 451: Solutions to Assignment #1 2.1) By definition, 2 Ω is the set of all subsets of Ω. Therefore, to show that 2 Ω is a σ-algebra we must show that the conditions of the definition σ-algebra are

More information

A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators

A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators Shiri Artstein-Avidan, Mathematics Department, Princeton University Abstract: In this paper we first

More information

Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide

Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide aliprantis.tex May 10, 2011 Aliprantis, Border: Infinite-dimensional Analysis A Hitchhiker s Guide Notes from [AB2]. 1 Odds and Ends 2 Topology 2.1 Topological spaces Example. (2.2) A semimetric = triangle

More information

Lower Tail Probabilities and Related Problems

Lower Tail Probabilities and Related Problems Lower Tail Probabilities and Related Problems Qi-Man Shao National University of Singapore and University of Oregon qmshao@darkwing.uoregon.edu . Lower Tail Probabilities Let {X t, t T } be a real valued

More information

Multiple points of the Brownian sheet in critical dimensions

Multiple points of the Brownian sheet in critical dimensions Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang Ecole Polytechnique Fédérale de Lausanne Based on joint work with: Carl Mueller Multiple points of the Brownian sheet in critical

More information

ITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES

ITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XL 2002 ITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES by Joanna Jaroszewska Abstract. We study the asymptotic behaviour

More information

A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM. MSC 2000: 03E05, 03E20, 06A10 Keywords: Chain Conditions, Boolean Algebras.

A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM. MSC 2000: 03E05, 03E20, 06A10 Keywords: Chain Conditions, Boolean Algebras. A BOREL SOLUTION TO THE HORN-TARSKI PROBLEM STEVO TODORCEVIC Abstract. We describe a Borel poset satisfying the σ-finite chain condition but failing to satisfy the σ-bounded chain condition. MSC 2000:

More information

Subdifferential representation of convex functions: refinements and applications

Subdifferential representation of convex functions: refinements and applications Subdifferential representation of convex functions: refinements and applications Joël Benoist & Aris Daniilidis Abstract Every lower semicontinuous convex function can be represented through its subdifferential

More information

Set, functions and Euclidean space. Seungjin Han

Set, functions and Euclidean space. Seungjin Han Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,

More information

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze

More information

P i [B k ] = lim. n=1 p(n) ii <. n=1. V i :=

P i [B k ] = lim. n=1 p(n) ii <. n=1. V i := 2.7. Recurrence and transience Consider a Markov chain {X n : n N 0 } on state space E with transition matrix P. Definition 2.7.1. A state i E is called recurrent if P i [X n = i for infinitely many n]

More information

The Lebesgue Integral

The Lebesgue Integral The Lebesgue Integral Brent Nelson In these notes we give an introduction to the Lebesgue integral, assuming only a knowledge of metric spaces and the iemann integral. For more details see [1, Chapters

More information

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3 Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

More information

2.2 Some Consequences of the Completeness Axiom

2.2 Some Consequences of the Completeness Axiom 60 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.2 Some Consequences of the Completeness Axiom In this section, we use the fact that R is complete to establish some important results. First, we will prove that

More information

Local semiconvexity of Kantorovich potentials on non-compact manifolds

Local semiconvexity of Kantorovich potentials on non-compact manifolds Local semiconvexity of Kantorovich potentials on non-compact manifolds Alessio Figalli, Nicola Gigli Abstract We prove that any Kantorovich potential for the cost function c = d / on a Riemannian manifold

More information

Introduction. Chapter 1. Contents. EECS 600 Function Space Methods in System Theory Lecture Notes J. Fessler 1.1

Introduction. Chapter 1. Contents. EECS 600 Function Space Methods in System Theory Lecture Notes J. Fessler 1.1 Chapter 1 Introduction Contents Motivation........................................................ 1.2 Applications (of optimization).............................................. 1.2 Main principles.....................................................

More information

Product metrics and boundedness

Product metrics and boundedness @ Applied General Topology c Universidad Politécnica de Valencia Volume 9, No. 1, 2008 pp. 133-142 Product metrics and boundedness Gerald Beer Abstract. This paper looks at some possible ways of equipping

More information

Inverse Brascamp-Lieb inequalities along the Heat equation

Inverse Brascamp-Lieb inequalities along the Heat equation Inverse Brascamp-Lieb inequalities along the Heat equation Franck Barthe and Dario Cordero-Erausquin October 8, 003 Abstract Adapting Borell s proof of Ehrhard s inequality for general sets, we provide

More information

A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS

A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS Fixed Point Theory, (0), No., 4-46 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS A. ABKAR AND M. ESLAMIAN Department of Mathematics,

More information

MULTIPLES OF HYPERCYCLIC OPERATORS. Catalin Badea, Sophie Grivaux & Vladimir Müller

MULTIPLES OF HYPERCYCLIC OPERATORS. Catalin Badea, Sophie Grivaux & Vladimir Müller MULTIPLES OF HYPERCYCLIC OPERATORS by Catalin Badea, Sophie Grivaux & Vladimir Müller Abstract. We give a negative answer to a question of Prajitura by showing that there exists an invertible bilateral

More information

The Polynomial Numerical Index of L p (µ)

The Polynomial Numerical Index of L p (µ) KYUNGPOOK Math. J. 53(2013), 117-124 http://dx.doi.org/10.5666/kmj.2013.53.1.117 The Polynomial Numerical Index of L p (µ) Sung Guen Kim Department of Mathematics, Kyungpook National University, Daegu

More information

An example of a convex body without symmetric projections.

An example of a convex body without symmetric projections. An example of a convex body without symmetric projections. E. D. Gluskin A. E. Litvak N. Tomczak-Jaegermann Abstract Many crucial results of the asymptotic theory of symmetric convex bodies were extended

More information

ANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS Mathematics Subject Classification: 11B05, 11B13, 11P99

ANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS Mathematics Subject Classification: 11B05, 11B13, 11P99 ANSWER TO A QUESTION BY BURR AND ERDŐS ON RESTRICTED ADDITION, AND RELATED RESULTS N. HEGYVÁRI, F. HENNECART AND A. PLAGNE Abstract. We study the gaps in the sequence of sums of h pairwise distinct elements

More information

Metric Spaces and Topology

Metric Spaces and Topology Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies

More information

Relationships between upper exhausters and the basic subdifferential in variational analysis

Relationships between upper exhausters and the basic subdifferential in variational analysis J. Math. Anal. Appl. 334 (2007) 261 272 www.elsevier.com/locate/jmaa Relationships between upper exhausters and the basic subdifferential in variational analysis Vera Roshchina City University of Hong

More information

Functional Analysis HW #3

Functional Analysis HW #3 Functional Analysis HW #3 Sangchul Lee October 26, 2015 1 Solutions Exercise 2.1. Let D = { f C([0, 1]) : f C([0, 1])} and define f d = f + f. Show that D is a Banach algebra and that the Gelfand transform

More information

The Sherrington-Kirkpatrick model

The Sherrington-Kirkpatrick model Stat 36 Stochastic Processes on Graphs The Sherrington-Kirkpatrick model Andrea Montanari Lecture - 4/-4/00 The Sherrington-Kirkpatrick (SK) model was introduced by David Sherrington and Scott Kirkpatrick

More information

Notes on Gaussian processes and majorizing measures

Notes on Gaussian processes and majorizing measures Notes on Gaussian processes and majorizing measures James R. Lee 1 Gaussian processes Consider a Gaussian process {X t } for some index set T. This is a collection of jointly Gaussian random variables,

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

Duality of multiparameter Hardy spaces H p on spaces of homogeneous type

Duality of multiparameter Hardy spaces H p on spaces of homogeneous type Duality of multiparameter Hardy spaces H p on spaces of homogeneous type Yongsheng Han, Ji Li, and Guozhen Lu Department of Mathematics Vanderbilt University Nashville, TN Internet Analysis Seminar 2012

More information

STAT 7032 Probability Spring Wlodek Bryc

STAT 7032 Probability Spring Wlodek Bryc STAT 7032 Probability Spring 2018 Wlodek Bryc Created: Friday, Jan 2, 2014 Revised for Spring 2018 Printed: January 9, 2018 File: Grad-Prob-2018.TEX Department of Mathematical Sciences, University of Cincinnati,

More information

Walker Ray Econ 204 Problem Set 2 Suggested Solutions July 22, 2017

Walker Ray Econ 204 Problem Set 2 Suggested Solutions July 22, 2017 Walker Ray Econ 204 Problem Set 2 Suggested s July 22, 2017 Problem 1. Show that any set in a metric space (X, d) can be written as the intersection of open sets. Take any subset A X and define C = x A

More information

Angle contraction between geodesics

Angle contraction between geodesics Angle contraction between geodesics arxiv:0902.0315v1 [math.ds] 2 Feb 2009 Nikolai A. Krylov and Edwin L. Rogers Abstract We consider here a generalization of a well known discrete dynamical system produced

More information

Empirical Processes and random projections

Empirical Processes and random projections Empirical Processes and random projections B. Klartag, S. Mendelson School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA. Institute of Advanced Studies, The Australian National

More information

Examples of Dual Spaces from Measure Theory

Examples of Dual Spaces from Measure Theory Chapter 9 Examples of Dual Spaces from Measure Theory We have seen that L (, A, µ) is a Banach space for any measure space (, A, µ). We will extend that concept in the following section to identify an

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

SELECTIVELY BALANCING UNIT VECTORS AART BLOKHUIS AND HAO CHEN

SELECTIVELY BALANCING UNIT VECTORS AART BLOKHUIS AND HAO CHEN SELECTIVELY BALANCING UNIT VECTORS AART BLOKHUIS AND HAO CHEN Abstract. A set U of unit vectors is selectively balancing if one can find two disjoint subsets U + and U, not both empty, such that the Euclidean

More information

Convex inequalities, isoperimetry and spectral gap III

Convex inequalities, isoperimetry and spectral gap III Convex inequalities, isoperimetry and spectral gap III Jesús Bastero (Universidad de Zaragoza) CIDAMA Antequera, September 11, 2014 Part III. K-L-S spectral gap conjecture KLS estimate, through Milman's

More information

Topology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.

Topology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved. Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124

More information

Convex Geometry. Carsten Schütt

Convex Geometry. Carsten Schütt Convex Geometry Carsten Schütt November 25, 2006 2 Contents 0.1 Convex sets... 4 0.2 Separation.... 9 0.3 Extreme points..... 15 0.4 Blaschke selection principle... 18 0.5 Polytopes and polyhedra.... 23

More information

Sequence of maximal distance codes in graphs or other metric spaces

Sequence of maximal distance codes in graphs or other metric spaces Electronic Journal of Graph Theory and Applications 1 (2) (2013), 118 124 Sequence of maximal distance codes in graphs or other metric spaces C. Delorme University Paris Sud 91405 Orsay CEDEX - France.

More information

Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012

Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Definitions (2 points each) 1. State the definition of a metric space. A metric space (X, d) is set

More information

A note on the convex infimum convolution inequality

A note on the convex infimum convolution inequality A note on the convex infimum convolution inequality Naomi Feldheim, Arnaud Marsiglietti, Piotr Nayar, Jing Wang Abstract We characterize the symmetric measures which satisfy the one dimensional convex

More information

Nonamenable Products are not Treeable

Nonamenable Products are not Treeable Version of 30 July 1999 Nonamenable Products are not Treeable by Robin Pemantle and Yuval Peres Abstract. Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism

More information

Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes with Killing

Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes with Killing Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 401 412 (2013) http://campus.mst.edu/adsa Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes

More information

Subspaces and orthogonal decompositions generated by bounded orthogonal systems

Subspaces and orthogonal decompositions generated by bounded orthogonal systems Subspaces and orthogonal decompositions generated by bounded orthogonal systems Olivier GUÉDON Shahar MENDELSON Alain PAJOR Nicole TOMCZAK-JAEGERMANN August 3, 006 Abstract We investigate properties of

More information

A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand

A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand Carl Mueller 1 and Zhixin Wu Abstract We give a new representation of fractional

More information

U e = E (U\E) e E e + U\E e. (1.6)

U e = E (U\E) e E e + U\E e. (1.6) 12 1 Lebesgue Measure 1.2 Lebesgue Measure In Section 1.1 we defined the exterior Lebesgue measure of every subset of R d. Unfortunately, a major disadvantage of exterior measure is that it does not satisfy

More information

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:

MATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous: MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is

More information

The Folk Theorem for Finitely Repeated Games with Mixed Strategies

The Folk Theorem for Finitely Repeated Games with Mixed Strategies The Folk Theorem for Finitely Repeated Games with Mixed Strategies Olivier Gossner February 1994 Revised Version Abstract This paper proves a Folk Theorem for finitely repeated games with mixed strategies.

More information

Boolean Algebras. Chapter 2

Boolean Algebras. Chapter 2 Chapter 2 Boolean Algebras Let X be an arbitrary set and let P(X) be the class of all subsets of X (the power set of X). Three natural set-theoretic operations on P(X) are the binary operations of union

More information

Existence and data dependence for multivalued weakly Ćirić-contractive operators

Existence and data dependence for multivalued weakly Ćirić-contractive operators Acta Univ. Sapientiae, Mathematica, 1, 2 (2009) 151 159 Existence and data dependence for multivalued weakly Ćirić-contractive operators Liliana Guran Babeş-Bolyai University, Department of Applied Mathematics,

More information

ZERO DUALITY GAP FOR CONVEX PROGRAMS: A GENERAL RESULT

ZERO DUALITY GAP FOR CONVEX PROGRAMS: A GENERAL RESULT ZERO DUALITY GAP FOR CONVEX PROGRAMS: A GENERAL RESULT EMIL ERNST AND MICHEL VOLLE Abstract. This article addresses a general criterion providing a zero duality gap for convex programs in the setting of

More information

Maths 212: Homework Solutions

Maths 212: Homework Solutions Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

More information

Supermodular ordering of Poisson arrays

Supermodular ordering of Poisson arrays Supermodular ordering of Poisson arrays Bünyamin Kızıldemir Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 637371 Singapore

More information

Some Background Material

Some Background Material Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

More information

Brownian Motion and Conditional Probability

Brownian Motion and Conditional Probability Math 561: Theory of Probability (Spring 2018) Week 10 Brownian Motion and Conditional Probability 10.1 Standard Brownian Motion (SBM) Brownian motion is a stochastic process with both practical and theoretical

More information

On the simplest expression of the perturbed Moore Penrose metric generalized inverse

On the simplest expression of the perturbed Moore Penrose metric generalized inverse Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated

More information

Renormings of c 0 and the minimal displacement problem

Renormings of c 0 and the minimal displacement problem doi: 0.55/umcsmath-205-0008 ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL. LXVIII, NO. 2, 204 SECTIO A 85 9 ŁUKASZ PIASECKI Renormings of c 0 and the minimal displacement problem Abstract.

More information

Extreme points of compact convex sets

Extreme points of compact convex sets Extreme points of compact convex sets In this chapter, we are going to show that compact convex sets are determined by a proper subset, the set of its extreme points. Let us start with the main definition.

More information

ON CONVERGENCE RATES OF GIBBS SAMPLERS FOR UNIFORM DISTRIBUTIONS

ON CONVERGENCE RATES OF GIBBS SAMPLERS FOR UNIFORM DISTRIBUTIONS The Annals of Applied Probability 1998, Vol. 8, No. 4, 1291 1302 ON CONVERGENCE RATES OF GIBBS SAMPLERS FOR UNIFORM DISTRIBUTIONS By Gareth O. Roberts 1 and Jeffrey S. Rosenthal 2 University of Cambridge

More information

Necessary and Sufficient Conditions for the Existence of a Global Maximum for Convex Functions in Reflexive Banach Spaces

Necessary and Sufficient Conditions for the Existence of a Global Maximum for Convex Functions in Reflexive Banach Spaces Laboratoire d Arithmétique, Calcul formel et d Optimisation UMR CNRS 6090 Necessary and Sufficient Conditions for the Existence of a Global Maximum for Convex Functions in Reflexive Banach Spaces Emil

More information

On Monch type multi-valued maps and fixed points

On Monch type multi-valued maps and fixed points Applied Mathematics Letters 20 (2007) 622 628 www.elsevier.com/locate/aml On Monch type multi-valued maps and fixed points B.C. Dhage Kasubai, Gurukul Colony, Ahmedpur-413 515 Dist: Latur, Maharashtra,

More information

Lecture 18: March 15

Lecture 18: March 15 CS71 Randomness & Computation Spring 018 Instructor: Alistair Sinclair Lecture 18: March 15 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They may

More information

On metric characterizations of some classes of Banach spaces

On metric characterizations of some classes of Banach spaces On metric characterizations of some classes of Banach spaces Mikhail I. Ostrovskii January 12, 2011 Abstract. The first part of the paper is devoted to metric characterizations of Banach spaces with no

More information

NOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS

NOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS NOWHERE LOCALLY UNIFORMLY CONTINUOUS FUNCTIONS K. Jarosz Southern Illinois University at Edwardsville, IL 606, and Bowling Green State University, OH 43403 kjarosz@siue.edu September, 995 Abstract. Suppose

More information

arxiv: v1 [math.pr] 11 Jan 2013

arxiv: v1 [math.pr] 11 Jan 2013 Last-Hitting Times and Williams Decomposition of the Bessel Process of Dimension 3 at its Ultimate Minimum arxiv:131.2527v1 [math.pr] 11 Jan 213 F. Thomas Bruss and Marc Yor Université Libre de Bruxelles

More information

UNIFORM EMBEDDINGS OF BOUNDED GEOMETRY SPACES INTO REFLEXIVE BANACH SPACE

UNIFORM EMBEDDINGS OF BOUNDED GEOMETRY SPACES INTO REFLEXIVE BANACH SPACE UNIFORM EMBEDDINGS OF BOUNDED GEOMETRY SPACES INTO REFLEXIVE BANACH SPACE NATHANIAL BROWN AND ERIK GUENTNER ABSTRACT. We show that every metric space with bounded geometry uniformly embeds into a direct

More information

On Reflecting Brownian Motion with Drift

On Reflecting Brownian Motion with Drift Proc. Symp. Stoch. Syst. Osaka, 25), ISCIE Kyoto, 26, 1-5) On Reflecting Brownian Motion with Drift Goran Peskir This version: 12 June 26 First version: 1 September 25 Research Report No. 3, 25, Probability

More information

Much of this Chapter is unedited or incomplete.

Much of this Chapter is unedited or incomplete. Chapter 9 Chaining methods Much of this Chapter is unedited or incomplete. 1. The chaining strategy The previous Chapter gave several ways to bound the tails of max i N X i. The chaining method applies

More information

arxiv: v1 [math.pr] 22 Dec 2018

arxiv: v1 [math.pr] 22 Dec 2018 arxiv:1812.09618v1 [math.pr] 22 Dec 2018 Operator norm upper bound for sub-gaussian tailed random matrices Eric Benhamou Jamal Atif Rida Laraki December 27, 2018 Abstract This paper investigates an upper

More information

Global Maximum of a Convex Function: Necessary and Sufficient Conditions

Global Maximum of a Convex Function: Necessary and Sufficient Conditions Journal of Convex Analysis Volume 13 2006), No. 3+4, 687 694 Global Maximum of a Convex Function: Necessary and Sufficient Conditions Emil Ernst Laboratoire de Modélisation en Mécaniue et Thermodynamiue,

More information

Analysis III. Exam 1

Analysis III. Exam 1 Analysis III Math 414 Spring 27 Professor Ben Richert Exam 1 Solutions Problem 1 Let X be the set of all continuous real valued functions on [, 1], and let ρ : X X R be the function ρ(f, g) = sup f g (1)

More information

MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1

MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1 MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION ELEMENTS OF SOLUTION Problem 1 1. Let X be a Hausdorff space and K 1, K 2 disjoint compact subsets of X. Prove that there exist disjoint open sets U 1 and

More information

Ordinary differential equations. Recall that we can solve a first-order linear inhomogeneous ordinary differential equation (ODE) dy dt

Ordinary differential equations. Recall that we can solve a first-order linear inhomogeneous ordinary differential equation (ODE) dy dt MATHEMATICS 3103 (Functional Analysis) YEAR 2012 2013, TERM 2 HANDOUT #1: WHAT IS FUNCTIONAL ANALYSIS? Elementary analysis mostly studies real-valued (or complex-valued) functions on the real line R or

More information

Continuous Sets and Non-Attaining Functionals in Reflexive Banach Spaces

Continuous Sets and Non-Attaining Functionals in Reflexive Banach Spaces Laboratoire d Arithmétique, Calcul formel et d Optimisation UMR CNRS 6090 Continuous Sets and Non-Attaining Functionals in Reflexive Banach Spaces Emil Ernst Michel Théra Rapport de recherche n 2004-04

More information

GRAPHS OF CONVEX FUNCTIONS ARE σ1-straight

GRAPHS OF CONVEX FUNCTIONS ARE σ1-straight ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 33, Number 3, Fall 2003 GRAPHS OF CONVEX FUNCTIONS ARE σ1-straight RICHARD DELAWARE ABSTRACT. A set E R n is s-straight for s>0ife has finite Method II outer

More information

AN INEQUALITY FOR TAIL PROBABILITIES OF MARTINGALES WITH BOUNDED DIFFERENCES

AN INEQUALITY FOR TAIL PROBABILITIES OF MARTINGALES WITH BOUNDED DIFFERENCES Lithuanian Mathematical Journal, Vol. 4, No. 3, 00 AN INEQUALITY FOR TAIL PROBABILITIES OF MARTINGALES WITH BOUNDED DIFFERENCES V. Bentkus Vilnius Institute of Mathematics and Informatics, Akademijos 4,

More information

Iowa State University. Instructor: Alex Roitershtein Summer Homework #1. Solutions

Iowa State University. Instructor: Alex Roitershtein Summer Homework #1. Solutions Math 501 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 015 EXERCISES FROM CHAPTER 1 Homework #1 Solutions The following version of the

More information

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory

Part V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite

More information

2. Transience and Recurrence

2. Transience and Recurrence Virtual Laboratories > 15. Markov Chains > 1 2 3 4 5 6 7 8 9 10 11 12 2. Transience and Recurrence The study of Markov chains, particularly the limiting behavior, depends critically on the random times

More information

A Geometric Interpretation of the Metropolis Hastings Algorithm

A Geometric Interpretation of the Metropolis Hastings Algorithm Statistical Science 2, Vol. 6, No., 5 9 A Geometric Interpretation of the Metropolis Hastings Algorithm Louis J. Billera and Persi Diaconis Abstract. The Metropolis Hastings algorithm transforms a given

More information

A note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series

A note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series Publ. Math. Debrecen 73/1-2 2008), 1 10 A note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series By SOO HAK SUNG Taejon), TIEN-CHUNG

More information

CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS

CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS Abstract. The aim of this paper is to characterize in terms of classical (quasi)convexity of extended real-valued functions the set-valued maps which are

More information